Here we state and prove a decay result in the case of equal wave speeds propagation.

Define the state spaces

where

The associated energy term is given by

By a straightforward calculation, we have

From semigroup theory [25, 26], we have the following existence and regularity result; for the explicit proofs, we refer the reader to [16].

Lemma 2.1.

Let be given. Then problem (1.1)–(1.5) has a unique global weak solution verifying

We are now ready to state our main stability result.

Theorem 2.2.

Suppose that and . Then the energy decays exponentially as time tends to infinity; that is, there exist two positive constants and *μ* independent of the initial data and , such that

The proof of our result will be established through several lemmas.

Let

where is the solution of

Lemma 2.3.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

By using the inequalities

and Young's inequality, the assertion of the lemma follows.

Let

Lemma 2.4.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

Using (1.4) and (1.1), we get

The assertion of the lemma then follows, using Young's and Poincaré's inequalities.

Let

Lemma 2.5.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

Using (1.3) and (1.5), we have

Then, using Young's and Poincaré's inequalities, we can obtain the assertion.

Next, we set

Lemma 2.6.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

Letting , , then using (1.2), (1.3), we have

Noticing that , then

Then, using Young's inequality, we can obtain the assertion.

We set

Lemma 2.7.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

Let , , then using (1.1), (1.2), we have

Then, noticing , again, from the above two equalities and Young's inequality, we can obtain the assertion.

Next, we set

Lemma 2.8.

Letting , , , , be a solution of (1.1)–(1.5), then one has

Proof.

Using (1.1), (1.2), we have

Noticing (2.3) and (2.4), we have that satisfy the following:

Similarly,

Then, insert (2.28) and (2.29) into (2.27), and the assertion of the lemma follows.

Now, we set

Lemma 2.9.

Letting , , , , be a solution of (1.1)–(1.5), then one has, for all ,

Proof.

Using (1.5), we have

Then, using Young's and Poincaré's inequalities, we can obtain the assertion.

Now, letting , we define the Lyapunov functional as follows:

By using (2.4), (2.9), (2.13), (2.16), (2.19), (2.23), (2.26), and (2.31), we have

where

We can choose big enough, small enough, and

Then are all negative constants; at this point, there exists a constant , and (2.34) takes the form

We are now ready to prove Theorem 2.2.

Proof of Theorem 2.2.

Firstly, from the definition of , we have

which, from (2.37) and (2.38), leads to

Integrating (2.39) over and using (2.38) lead to (2.6). This completes the proof of Theorem 2.2.